Mosunetuzumab and Polatuzumab Vedotin for the Treatment of Patients With Relapsed or Refractory Grade 1-3a Follicular Lymphoma

Inclusion Criteria: * Documented informed consent of the participant and/or legally authorized representative * Assent, when appropriate, will be obtained per institutional guidelines * Be willing to provide tissue from a fresh core or excisional biopsy (performed as standard of care) of a tumor lesion prior to starting study therapy or from diagnostic tumor biopsies * If unavailable, exceptions may be granted with study principal investigator (PI) approval * Age: ≥ 18 years * Eastern Cooperative Oncology Group (ECOG) ≤ 2 * Histologically confirmed diagnosis of follicular lymphoma grade 1-3a according to the World Health Organization (WHO) classification. Note: A history of diffuse large B-cell lymphoma (DLBCL) and/or grade 3b follicular lymphoma is allowed (follicular large B-cell lymphoma in the WHO 5th classification of mature B-cell lymphoma) but these cannot be present at the time of study enrollment. Enrollment of any such cases on this study must receive prior approval by the study PI * Relapsed/ refractory disease after at least one prior line of therapy. Relapse must have been confirmed histologically * Tumor must be positive for CD20 by immunohistochemistry or flow cytometry after the most recent therapy. Note: As a pan-B marker, CD79B+ is nearly always present and is not a criterion for inclusion * Active disease requiring treatment per treating physician's decision * Radiographically measurable disease by Lugano criteria (e.g., one or more nodal sites of disease ≥ 1.5 cm and/or at least one extranodal site of disease ≥ 1.0 cm in longest dimension) * Fully recovered from the acute toxic effects (except alopecia) to ≤ grade 1 to prior anti-cancer therapy * WITHOUT BONE MARROW INVOLVEMENT BY LYMPHOMA: Absolute neutrophil count (ANC) ≥ 1,000/mm\^3. NOTE: Growth factor is not permitted within 7 days of ANC assessment unless cytopenia is secondary to disease involvement * WITH BONE MARROW INVOLVEMENT BY LYMPHOMA AND/OR DISEASE-RELATED NEUTROPENIAS: ANC ≥ 500/mm\^3. NOTE: Growth factor is not permitted within 7 days of ANC assessment unless cytopenia is secondary to disease involvement * WITHOUT BONE MARROW INVOLVEMENT BY LYMPHOMA: Platelets ≥ 75,000/mm\^3. NOTE: Platelet transfusions are not permitted within 7 days of platelet assessment unless cytopenia is secondary to disease involvement * WITH BONE MARROW INVOLVEMENT BY LYMPHOMA AND/OR DISEASE-RELATED CYTOPENIAS: Platelets ≥ 50,000/mm\^3. NOTE: Platelet transfusions are not permitted within 7 days of platelet assessment unless cytopenia is secondary to disease involvement * Hemoglobin ≥ 8 g/dL. NOTE: Red blood cell transfusions are not permitted within 7 days of hemoglobin assessment unless cytopenia is secondary to disease involvement * Total bilirubin ≤ 1.5 x upper limit of normal (ULN). If hepatic involvement by lymphoma, or Gilbert's disease: ≤ 3 x ULN * Aspartate aminotransferase (AST) ≤ 2.5 x ULN. If hepatic involvement by lymphoma: AST ≤ 5 x ULN * Alanine aminotransferase (ALT) ≤ 2.5 x ULN. If hepatic involvement by lymphoma: ALT ≤ 5 x ULN * Creatinine clearance of ≥ 40 mL/min per 24 hour urine test or the Cockcroft-Gault formula * IF NOT RECEIVING ANTICOAGULANTS: International normalized ratio (INR) OR prothrombin (PT) ≤ 1.5 x ULN * IF ON ANTICOAGULANT THERAPY: PT must be within therapeutic range of intended use of anticoagulants * IF NOT RECEIVING ANTICOAGULANTS: Activated partial thromboplastin time (aPTT) ≤ 1.5 x ULN * IF ON ANTICOAGULANT THERAPY: aPTT must be within therapeutic range of intended use of anticoagulants * WOMEN OF CHILDBEARING POTENTIAL (WOCBP): Negative urine or serum pregnancy test. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required * Agreement by females of childbearing potential to abstain from heterosexual intercourse or use two adequate method of birth control, including at least 1 method with a failure rate of \< 1% per year, for at least 28 days prior to day 1 of cycle 1, during the treatment period (including periods of treatment interruption), until 3 months after the final dose mosunetuzumab, 3 months after the last dose of polatuzumab vedotin, and 3 months after the last dose of tocilizumab (if applicable). Women must refrain from donating eggs during this same period. Agreement by males to abstain from heterosexual intercourse or use a condom with female partners of childbearing potential or pregnant female partners during the treatment period and until 5 months after the last dose of polatuzumab vedotin, and 2 months after the last dose of tocilizumab (if applicable). Men must refrain from donating sperm during this same period * Childbearing potential defined as not being surgically sterilized (men and women) or have not been free from menses for \> 1 year (women only) with no identified cause other than menopause * Examples of contraceptive methods with a failure rate of \< 1% per year include bilateral tubal ligation, male sterilization, established proper use of hormonal contraceptives that inhibit ovulation, hormone releasing intrauterine devices, and copper intrauterine devices. The reliability of sexual abstinence should be evaluated in relation to the duration of the clinical trial and the preferred and usual lifestyle of the patient. Periodic abstinence (e.g., calendar, ovulation, symptothermal, or post-ovulation methods) and withdrawal are not acceptable methods of contraception Exclusion Criteria: * Prior treatment with mosunetuzumab or other CD20-directed bispecific antibodies. Prior exposure to 2 or less doses without evidence of resistance is allowed * Prior treatment with polatuzumab vedotin or with an antibody-drug conjugate containing monomethyl auristatin E (MMAE). Prior exposure to 2 or less doses without evidence of resistance is allowed * Allogeneic stem cell transplant within 2 years prior to day 1 of protocol therapy, requires immunosuppression, or has evidence of active-versus-host-disease * Patients who had an allogeneic transplant \> 2 years prior to day 1 of protocol therapy must additionally have been stable off immunosuppressive agents for ≥ 2 months * Autologous stem cell transplant within 100 days prior to day 1 of protocol therapy * Chimeric antigen receptor (CAR)-T therapy within 30 days prior to day 1 of protocol therapy * Prior use of any anti-lymphoma treatment with monoclonal antibody, radioimmunoconjugate or antineoplastic drug conjugate (ADC) within 4 weeks prior to day 1 of protocol therapy * Treatment with any chemotherapeutic agent, or treatment with any other anti-cancer agent (investigational or otherwise) within 4 weeks or 5 half-lives of the drug, whichever is shorter, prior to day 1 of protocol therapy * Treatment with radiotherapy within 2 weeks prior to day 1 of protocol therapy * If patients have received radiotherapy within 4 weeks prior to prior to day 1 of protocol therapy, patients must have at least one measurable lesion outside of the radiation field. Patients who have only one measurable lesion that was previously irradiated but subsequently progressed are eligible * Live vaccine within 30 days prior to day 1 of protocol therapy * Systemic immunosuppressive therapy (including, but not limited to, cyclophosphamide, azathioprine, methotrexate, thalidomide, and anti-tumor necrosis factor \[TNF\] agents) with the exception of corticosteroid treatment for lymphoma symptom control must be tapered down to ≤ 10 mg/day prednisone or equivalent 14 days prior to day 1 of protocol therapy. Exceptions are: * Inhaled or topical steroids * Use of mineralocorticoids for management of orthostatic hypotension * Use of physiologic doses of corticosteroids for management of adrenal insufficiency * Use of 4 or less pulsed doses of steroids (i.e. dexamethasone) for urgent stabilization of lymphoma or symptom management * Grade ≥ 2 peripheral neuropathy * History of severe allergic or anaphylactic reaction to humanized, chimeric or murine monoclonal antibodies (or recombinant antibody-related fusion proteins) * History of solid organ transplantation * History of progressive multifocal leukoencephalopathy (PML) * Known or suspected history of hemophagocytic lymphohistiocytosis (HLH) * Significant active pulmonary disease (e.g., bronchospasm and/or obstructive pulmonary disease) * Pneumonitis or interstitial lung disease requiring ongoing corticosteroid or immunosuppression and/or requirement for supplemental oxygen * Patients with any other history of pneumonitis or interstitial lung disease may be eligible after discussion with the study PI * Positive severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) antigen or polymerase chain reaction (PCR) test within 7 days prior to day 1 of protocol therapy * Clinically significant uncontrolled illness * Known active bacterial, viral, fungal, mycobacterial, parasitic, or other infection (excluding fungal infections of nail beds) at study enrollment or any major episode of infection requiring treatment with IV antibiotics or hospitalization (relating to the completion of the course of antibiotics) within 2 weeks prior to the first study treatment administration * Known active hepatitis B virus (HBV) or hepatitis C virus (HCV) infection. Patients with past HBV infection (defined as negative hepatitis B surface antigen \[HBsAg\] and positive hepatitis B core antibody \[HBcAb\]) are eligible if HBV deoxyribonucleic acid (DNA) is undetectable. Patients who are positive for HCV antibody are eligible if PCR is negative for HCV ribonucleic acid (RNA). Testing to be done only in patients suspected of having infections or exposures * Known active human immunodeficiency virus (HIV) infection. Subjects who have an undetectable or unquantifiable HIV viral load with CD4 \> 200 and are on highly active antiretroviral therapy (HAART) medication are allowed. Testing to be done only in patients suspected of having infections or exposures * Known or suspected chronic active Epstein-Barr virus (EBV) infection * Known active central nervous system (CNS) involvement by lymphoma, including leptomeningeal involvement * Previously treated CNS involvement including disease with leptomeningeal involvement, is acceptable. Patients should be neurologically stable prior to study entry, and receiving a stable or decreasing corticosteroid dose * History of CNS disease which was symptomatic or required treatment in the past 1 year, such as stroke, epilepsy, CNS vasculitis or neurodegenerative disease * Symptomatic cardiac disease such as New York Heart Association class III or IV (including symptomatic ventricular dysfunction, symptomatic coronary artery disease, and symptomatic arrhythmias), cerebrovascular event/stroke or myocardial infarction within the past 6 months * Clinically significant history of liver disease, including viral or other hepatitis, or cirrhosis * Active autoimmune disease requiring systemic treatment * History of autoimmune disease, including, but not limited to, myasthenia gravis, myositis, autoimmune hepatitis, inflammatory bowel disease, vascular thrombosis associated with antiphospholipid syndrome, Wegener granulomatosis, Guillain-Barré syndrome, multiple sclerosis, vasculitis, or glomerulonephritis * Patients with a history of autoimmune-related hypothyroidism on a stable dose of thyroid replacement hormone may be eligible * Patients with controlled type 1 diabetes mellitus who are on an insulin regimen are eligible for the study * Patients with a history of rheumatoid arthritis, Sjögren syndrome, systemic lupus erythematosus may be eligible after discussion with the study PI * Patients with a history of disease-related immune thrombocytopenic purpura, autoimmune hemolytic anemia, or other stable autoimmune diseases may be eligible after discussion with the study PI * Recent major surgery (within 4 weeks) prior to start of protocol therapy, other than for diagnosis * History of another primary malignancy that has not been in remission for at least 2 years, with the following exceptions: * Adequately treated non-melanoma skin cancer or lentigo maligna (melanoma in situ) without evidence of disease * Adequately treated in situ carcinomas (e.g. cervical, esophageal) without evidence of disease * Asymptomatic prostate cancer managed with a watch-and-wait strategy * If the malignancy is expected to not require any treatment for at least 2 years (this exception should be discussed with the study PI) * Concomitant investigational therapeutic therapy or within 7 days prior to initiation of study treatment * FEMALES ONLY: Pregnant or breastfeeding * Any other condition that would, in the investigator's judgment, contraindicate the patient's participation in the clinical study due to safety concerns with clinical study procedures * Prospective participants who, in the opinion of the investigator, may not be able to comply with all study procedures (including compliance issues related to feasibility/logistics)